After an entertaining first-half, the visitors went ahead when Maximiliano Urruti stuck across goal and into the bottom corner.
Dave Romney won his side a penalty just six minutes later and Giovani dos Santos levelled the game by firing the spot kick home.
The scoring wasn’t done there, Acosta caught the ‘keeper out at his near post on 69 minutes and Dallas went on to hold out for their first win away against the Galaxy since 2009.
The Toros were in action through the week in the CONCACAF Champions League – beating Arabe Unido 5-2 on aggregate – and were looking to start the new MLS season with a win.
Sides can't be separated in entertaining first half
Dallas almost went ahead inside the opening three minutes when Cristian Colmán latched onto a through ball over the top from Matt Hedges.
The forward cut back inside the area, but Galaxy ‘keeper Brian Rowe was quick to close down and sweep away the danger.
Dallas came within inches of breaking the deadlock shortly after the half-hour mark when the ball broke to Michael Barrios at the back of the box.
The midfielder controlled Roland Lamah’s cross and stroked the ball under the ‘keeper, but Jelle Van Damme was on the line to scramble the ball clear.
Dallas break deadlock two minutes after half-time
Óscar Pareja’s side did manage to find the first goal in the game just two minutes after the break when Urruti made the most of the space he was given.
The Galaxy defenders dropped off the forward, who drove with the ball, struck across goal and the ball nestled into the bottom corner.
However, Dallas’ lead didn’t last long because Los Angeles found themselves with a great chance to level the game just six minutes later.
Romney was brought down in the area by Hernán Grana and Giaoani stepped up to the mark and fired the penalty into the top corner.
Jermaine Jones rose above three Dallas players at the back post and nodded Giovani’s corner onto the foot of the post before the ball was scrambled clear.
Acosta nets winner minutes after LA strike post
The visitors made the most of that slice of luck and managed to go ahead for the second time in the game five minutes later.
Lamah threaded the ball through to Acosta in the area before the midfielder quickly took his shot and caught the ‘keeper out at his near post.
The home side closed out the game strongly and could have drawn level once more, but substitute Ariel Lassiter couldn’t direct his header towards goal.
A long throw-in was knocked down into the path of Lassiter in the fourth minute of additional time, and his drilled effort took a slight deflection and sailed just over the bar.